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<h1 class="center">Sets and Venn Diagrams</h1>
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<h2>Sets</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/set-clothes.svg" alt="set of clothes" height="153" width="244"></p>
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<p>A <a href="sets-introduction.html">set</a> is a collection of things.</p>
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<p>For example, the items you wear is a set: these include hat, shirt, jacket, pants, and so on.</p>
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<p>You write sets inside <b>curly brackets</b> like this:</p>
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<p class="center"><b>{hat, shirt, jacket, pants, ...}</b></p>
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<p>You can also have sets of numbers:</p>
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<ul>
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<li>Set of <a href="../whole-numbers.html">whole numbers</a>: {0, 1, 2, 3, ...}</li>
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<li>Set of <a href="../prime_numbers.html">prime numbers</a>: {2, 3, 5, 7, 11, 13, 17, ...}</li>
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</ul>
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<h2>Ten Best Friends</h2>
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<p>You could have a set made up of your ten best friends:</p>
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<ul>
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<li>{alex, blair, casey, drew, erin, francis, glen, hunter, ira, jade}</li>
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</ul>
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<p>Each friend is an "element" (or "member") of the set. It is normal to use <b>lowercase letters</b> for them.</p>
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<p> </p>
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<p><img src="../images/soccer-teams.jpg" alt="soccer teams" style="float:right; margin: 10px;" height="86" width="209"></p>
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<p>Now let's say that alex, casey, drew and hunter play <b>Soccer</b>:</p>
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<p class="center larger">Soccer = {alex, casey, drew, hunter}</p>
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<p><i>(It says the Set "Soccer" is made up of the
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elements alex, casey, drew and hunter.)</i></p>
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<p> </p>
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<p><img src="images/tennis.jpg" alt="tennis" style="float:right; margin: 10px;" height="105" width="200"></p>
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<p>And casey, drew and jade play <b>Tennis</b>:</p>
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<p class="center larger">Tennis = {casey, drew, jade}</p>
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<p align="left"> </p>
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<p align="left">We can put their names in two separate circles:</p>
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<p class="center"><img src="images/venn-separate.svg" alt="Soccer and Tennis Sets" height="133" width="285"></p>
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<h2>Union</h2>
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<p>You can now list your friends that play <b>Soccer OR Tennis</b>.</p>
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<p>This is called a "Union" of sets and has the special symbol <b>∪</b>:</p>
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<p class="center larger">Soccer <b>∪</b> Tennis = {alex, casey, drew, hunter, jade}</p>
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<p>Not everyone is in that set ... only your friends that play Soccer or Tennis (or both).</p>
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<p>In other words we combine the elements of the two sets.</p>
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<p>We can show that in a "Venn Diagram":</p>
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<p class="center"><img src="images/venn-1.svg" alt="Soccer and Tennis Sets Union" height="133" width="212"><br>
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Venn Diagram: Union of 2 Sets</p>
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<p>A Venn Diagram is clever because it shows lots of information:</p>
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<div class="bigul">
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<ul>
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<li>Do you see that alex, casey, drew and hunter are in the "Soccer" set?</li>
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<li>And that casey, drew and jade are in the "Tennis" set?</li>
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<li>And here is the clever thing: <b>casey and drew are in BOTH sets!</b></li>
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</ul>
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<p>All that in one small diagram.</p>
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</div>
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<h2>Intersection</h2>
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<p>"Intersection" is when you must be in BOTH sets.</p>
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<p>In our case that means <b>they play both Soccer AND Tennis</b> ... which is casey and drew.</p>
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<p>The special symbol for Intersection is an upside down "U" like this: <b>∩</b></p>
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<p>And this is how we write it:</p>
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<p class="center larger">Soccer <b>∩</b> Tennis = {casey, drew}</p>
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<p>In a Venn Diagram:</p>
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<p class="center"><img src="images/venn-2.svg" alt="Soccer and Tennis Sets Intersection" height="" width=""><br>
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Venn Diagram: Intersection of 2 Sets</p>
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<div class="fun">
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<h3>Which Way Does That "U" Go?</h3>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../data/images/union-cup.jpg" alt="union symbol looks like cup" height="88" width="102"></p>
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<p>Think of them as "cups": <span class="large">∪</span> holds more water than <span class="large">∩</span>, right?</p>
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<p>So Union <span class="large">∪</span> is the one with more elements than Intersection ∩</p>
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</div>
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<h2>Difference</h2>
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<p>You can also "subtract" one set from another.</p>
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<p>For example, taking Soccer and subtracting Tennis means people that <b> play Soccer but NOT Tennis</b> ... which is alex and hunter.</p>
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<p>And this is how we write it:</p>
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<p class="center larger">Soccer <b>−</b> Tennis = {alex, hunter}</p>
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<p>In a Venn Diagram:</p>
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<p class="center"><img src="images/venn-diff.svg" alt="Soccer and Tennis Sets Difference" height="134" width="212"><br>
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Venn Diagram: Difference of 2 Sets</p>
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<h2>Summary So Far</h2>
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<ul>
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<div class="bigul">
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<li><b class="larger">∪</b> is Union: is in either set or both sets</li>
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<li><b>∩</b> is Intersection: only in both sets</li>
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<li><b>−</b> is Difference: in one set but not the other</li>
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</div>
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</ul>
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<h2>Three Sets</h2>
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<p>You can also use Venn Diagrams for 3 sets.</p>
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<p>Let us say the third set is "Volleyball", which drew, glen and jade play:</p>
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<p class="center"><span class="larger">Volleyball = {drew, glen, jade}</span></p>
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<p>But let's be more "mathematical" and use a Capital Letter for each set:</p>
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<ul>
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<li><b>S</b> means the set of Soccer players</li>
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<li><b>T</b> means the set of Tennis players</li>
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<li><b>V</b> means the set of Volleyball players</li>
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</ul>
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<p>The Venn Diagram is now like this:</p>
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<p class="center"><img src="images/venn-3.svg" alt="Soccer, Tennis and Volleyball Sets Union" height="200" width="212"></p>
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<p class="center">Union of 3 Sets: S <b>∪</b> T <b>∪</b> V</p>
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<p>You can see (for example) that:</p>
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<ul>
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<li>drew plays Soccer, Tennis <b>and</b> Volleyball</li>
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<li>jade plays Tennis and Volleyball</li>
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<li>alex and hunter play Soccer, but don't play Tennis or Volleyball</li>
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<li>no-one plays <b>only</b> Tennis</li>
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</ul>
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<p>We can now have some fun with Unions and Intersections ...</p>
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<p class="center"><img src="images/venn-4.svg" alt="Soccer, Tennis and Volleyball Sets" height="199" width="212"><br>
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This is just the set S</p>
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<p class="center larger">S = {alex, casey, drew, hunter}</p>
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<p> </p>
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<p class="center"><img src="images/venn-5.svg" alt="Soccer, Tennis and Volleyball Sets Union of Tennis and Volleyball" height="199" width="212"><br>
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This is the Union of Sets T and V</p>
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<p class="center larger">T <b>∪</b> V = {casey, drew, jade, glen}</p>
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<p> </p>
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<p class="center"><img src="images/venn-6.svg" alt="Soccer, Tennis and Volleyball Sets Intersection of Soccer and Volleyball" height="199" width="212"><br>
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This is the <b>Intersection</b> of Sets S and V</p>
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<p class="center larger">S <b>∩</b> V = {drew}</p>
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<p>And how about this ...</p>
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<ul>
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<li>take the <b>previous set</b> S <b>∩</b> V</li>
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<li>then <b>subtract T</b>:</li>
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</ul>
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<p class="center"><img src="images/venn-7.svg" alt="Soccer, Tennis and Volleyball Sets" height="199" width="212"><br>
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This is the Intersection of Sets S and V <b>minus</b> Set T</p>
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<p class="center larger">(S <b>∩</b> V) <b>−</b> T = {}</p>
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<p>Hey, there is nothing there!</p>
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<p>That is OK, it is just the "Empty Set". It is still a set, so we use the curly brackets with nothing inside: {}</p>
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<p class="words">The <b>Empty Set</b> has no elements: {}</p>
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<h2>Universal Set</h2>
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<p class="words">The <b>Universal Set</b> is the set that has everything. Well, not <i>exactly</i> everything. <b>Everything that we are interested in now.</b></p>
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<div class="center80">
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<p>Sadly, the symbol is the letter "U" ... which is easy to confuse with the <b>∪</b> for Union. You just have to be careful, OK?</p>
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</div>
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<p>In our case the Universal Set is our Ten Best Friends.</p>
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<p class="center larger">U = {alex, blair, casey, drew, erin, francis, glen, hunter, ira, jade}</p>
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<p>We can show the Universal Set in a Venn Diagram by putting a box around the whole thing:</p>
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<p class="center"><img src="images/venn-8.svg" alt="Soccer, Tennis and Volleyball Sets" height="234" width="311"></p>
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<p>Now you can see ALL your ten best friends, neatly sorted into what sport they play (or not!).</p>
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<p>And then we can do interesting things like take the whole set and <b>subtract the ones who play Soccer</b>:</p>
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<p class="center"><img src="images/venn-9.svg" alt="Soccer, Tennis and Volleyball Sets" height="234" width="311"></p>
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<p>We write it this way:</p>
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<p class="center larger">U <b>−</b> S = {blair, erin, francis, glen, ira, jade}</p>
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<p>Which says "The Universal Set minus the Soccer Set is the Set {blair, erin, francis, glen, ira, jade}"</p>
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<p>In other words "everyone who does <b>not</b> play Soccer".</p>
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<h2>Complement</h2>
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<p>And there is a special way of saying "everything that is <b>not</b>", and it is called <i><b>"complement"</b></i>.</p>
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<p>We show it by writing a little "C" like this:</p>
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<p class="center large">S<sup>c</sup></p>
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<p>Which means "everything that is NOT in S", like this:</p>
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<p class="center"><img src="images/venn-9.svg" alt="Soccer, Tennis and Volleyball Sets" height="234" width="311"></p>
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<p class="center"><span class="larger">S<sup>c</sup> = {blair, erin, francis, glen, ira, jade}</span><br>
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(exactly the same as the <b>U − S</b> example from above)</p>
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<p> </p>
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<h2>Summary</h2>
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<ul>
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<div class="bigul">
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<li><b class="large">∪</b> is Union: is in either set or both sets</li>
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<li><b>∩</b> is Intersection: only in both sets</li>
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<li><b>−</b> is Difference: in one set but not the other</li>
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<li>A<sup>c</sup> is the Complement of A: everything that is not in A</li>
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<li>Empty Set: the set with no elements. Shown by {}</li>
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<li>Universal Set: all things we are interested in</li>
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</div>
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</ul>
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<p> </p>
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<div class="questions">1886, 1887, 1890, 1892, 7220, 1888, 1889, 1891, 91, 73</div>
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<div class="related">
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<a href="sets-introduction.html">Introduction to Sets</a>
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<a href="boolean-algebra.html">Boolean Algebra</a>
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<a href="logic-gates.html">Logic Gates</a>
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<a href="index.html">Sets Index</a>
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<div id="copyrt">Copyright © 2021 Rod Pierce</div>
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